As a new poll suggests, the increasingly stark ideological divides of American politics have come with personal consequences. Nearly one in five voters — 19 percent — said that politics had hurt their friendships or family relationships, according to a poll conducted last week by The New York Times and Siena College. For all the concern over violent political rhetoric and outright political conflict in the United States, the ruptures that people described were typically quiet ones made more in sorrow than anger, as people with years of common experience came to the conclusion that they no longer even agreed on enough facts to have coherent arguments. “There’s a great deal of hurt,” said Paul Lucky, 73, a child-care provider and a self-described left-leaning Democrat who lives near Sacramento and who spoke of a strained relationship with his Republican son. Close to half of the voters in the survey also acknowledged making judgments about other people based on their politics. Forty-eight percent of those polled said that knowing a person’s political views told them either a lot or a little about whether someone was a good person.